We see that Eq. (REF ) and Eq. (REF ) have
a similar structure. Indeed, in the good decoherence limit and for
pure conditional states, the accessible information, which is equivalent
to Eq. (REF ) or Eq. (REF ), is equal
to \(H_{\mathcal {S}}-h\left(P_{e}\right)\) . Here, \(P_{e}=\frac{1}{2}\left(1-\mathrm {tr}\left|p_{1}\rho _{\left.\mathcal {F}\right|1}-p_{2}\rho _{\left.\mathcal {F}\right|2}\right|\right)\) 
is the optimal error probability, which is given by the Helstrom measurement
[1]}, for single shot state discrimination
of the conditional fragment states [2]}, [3]}, [4]}.
This is not true for mixed or for higher dimensional pointer subspaces
[5]}, [6]}, [7]}, [8]}.
It can be verified by a direct computation of the error probability
from the optimal measurement for the pure conditional states. For
\(\rho _{\mathcal {F}\left|\hat{s}\right.}\)  pure, the trace distance in the Helstrom expression just requires
diagonalizing an operator in a two-dimensional subspace, giving
\(P_{e}=\frac{1}{2}\left(1-\sqrt{1-4p_{1}p_{2}\left|\gamma \right|^{2{^{\sharp }}\hspace{-1.99168pt}\mathcal {F}}}\right)\) 
